Abstract
In the edgedisjoint paths problem, we are given a graph and a set of k pairs of vertices, and we have to decide whether or not the graph has k edgedisjoint paths connecting given pairs of terminals. Robertson and Seymour’s graph minor project gives rise to a polynomial time algorithm for this problem for any fixed k, but their proof of the correctness needs the whole Graph Minor project. We give a faster algorithm and a much simpler proof of the correctness for the edgedisjoint paths problem. Our results can be summarized as follows:

1.
If an input graph is either 4edgeconnected or Eulerian, then our algorithm only needs to look for the following three simple reductions: (i) Excluding vertices of high degree. (ii) Excluding ≤3edgecuts. (iii) Excluding large clique minors.

2.
When an input graph is either 4edgeconnected or Eulerian, the number of terminals k is allowed to be a nontrivially super constant number, up to k=O((log log logn)^{½−ε}) for any ε > 0. In addition, if an input graph is either 4edgeconnected planar or Eulerian planar, k is allowed to be O((logn ^{½−ε}) for any ε > 0.

3.
We also give our own algorithm for the edgedisjoint paths problem in general graphs. We basically follow the RobertsonSeymour’s algorithm, but we cut half of the proof of the correctness for their algorithm. In addition, our algorithm is faster than Robertson and Seymour’s.
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References
 [1]
M. Andrews, J. Chuzhoy, S. Khanna and L. Zhang: Hardness of the undirected edgedisjoint paths problem with congestion, Proc. 46th IEEE Symposium on Foundations of Computer Science (FOCS), 2005, 226–244.
 [2]
S. Arnborg and A. Proskurowski: Linear time algorithms for NPhard problems restricted to partial ktrees, Discrete Appl. Math. 23 (1989), 11–24.
 [3]
H. L. Bodlaender: A lineartime algorithm for finding treedecomposition of small treewidth, SIAM J. Comput. 25 (1996), 1305–1317.
 [4]
C. Chekuri, S. Khanna and B. Shepherd: Edgedisjoint paths in planar graphs, Proc. 45th IEEE Symposium on Foundations of Computer Science (FOCS), 2004, 71–80.
 [5]
C. Chekuri, S. Khanna and B. Shepherd: Edgedisjoint paths in planar graphs with constant congestion, SIAM J. Comput. 39 (2009), 281–301.
 [6]
E.D. Demaine and M. Hajiaghayi: Fast algorithms for hard graph problems: Bidimensionality, minors, and local treewidth, Proc. 12th Internat. Symp. on Graph Drawing, Lecture Notes in Computer Science 3383, Springer, 2004, 517–533.
 [7]
E.D. Demaine and M. Hajiaghayi: Linearity of grid minors in treewidth with applications through bidimensionality, Combinatorica 28 (2008), 19–36.
 [8]
R. Diestel, K.Y. Gorbunov, T.R. Jensen and C. Thomassen: Highly connected sets and the excluded grid theorem, J. Combin. Theory Ser. B 75 (1999), 61–73.
 [9]
A. Frank: Packing paths, cuts and circuits a survey, in: Paths, Flows and VLSILayout, B. Korte, L. Lovász, H.J. Promel and A. Schrijver (Eds.), SpringerVerlag, Berlin, 1990, 49–100.
 [10]
V. Guruswami, S. Khanna, R. Rajaraman, B. Shepherd and M. Yannakakis: Nearoptimal hardness results and approximation algorithms for edgedisjoint paths and related problems, J. Comp. Styst. Sciences 67 (2003), 473–496.
 [11]
R. Halin: Sfunctions for graphs, J. Geometry 8 (1976), 171–176.
 [12]
D. Johnson: The many faces of polynomial time, in: The NPcompleteness column: An ongoing guide, J. Algorithms 8 (1987), 285–303.
 [13]
R. M. Karp: On the computational complexity of combinatorial problems, Networks 5 (1975), 45–68.
 [14]
K. Kawarabayashi and Y. Kobayashi: The edge disjoint paths problem in Eulerian graphs and 4edgeconnected graphs, Proc. 21st Annual ACMSIAM Symposium on Discrete Algorithms (SODA), 2010, 345–353.
 [15]
K. Kawarabayashi, Y. Kobayashi and B. Reed: The disjoint paths problem in quadratic time, J. Combin. Theory Ser. B 102 (2012), 424–435.
 [16]
K. Kawarabayashi and P. Wollan: A shorter proof of the graph minor algorithm the unique linkage theorem, Proc. 42nd ACM Symposium on Theory of Computing (STOC), 2010, 687–694.
 [17]
J. Kleinberg: An approximation algorithm for the disjoint paths problem in evendegree planar graphs, Proc. 46th IEEE Symposium on Foundations of Computer Science (FOCS), 2005, 627–636.
 [18]
J. Kleinberg and É. Tardos: Disjoint paths in densely embedded graphs, Proc. 36th IEEE Symposium on Foundations of Computer Science (FOCS), 1995, 52–61.
 [19]
J. Kleinberg and É. Tardos: Approximations for the disjoint paths problem in highdiameter planar networks, Proc. 27th ACM Symposium on Theory of Computing (STOC), 1995, 26–35.
 [20]
M. R. Kramer and J. van Leeuwen: The complexity of wirerouting and fingding minimum area layouts for arbitrary VLSI circuits, Adv. Comput. Res. 2 (1984), 129–146.
 [21]
M. Middendorf and F. Pfeiffer: On the complexity of the disjoint paths problem, Combinatorica 13 (1993), 97–107.
 [22]
H. Nagamochi and T. Ibaraki: A lineartime algorithm for finding a sparse kconnected spanning subgraph of a kconnected graph, Algorithmica 7 (1992), 583–596.
 [23]
T. Nishizeki, J. Vygen and X. Zhou: The edgedisjoint paths problem is NPcomplete for seriesparallel graphs, Discrete Appl. Math. 115 (2001), 177–186.
 [24]
H. Okamura and P. D. Seymour: Multicommodity ows in planar graphs. J. Combin. Theory Ser. B 31 (1981), 75–81.
 [25]
L. Perkovic and B. Reed: An improved algorithm for finding tree decompositions of small width, International Journal on the Foundations of Computing Science 11 (2000), 81–85.
 [26]
B. Reed: Finding approximate separators and computing tree width quickly, Proc. 24th ACM Symposium on Theory of Computing (STOC), 1992, 221–228.
 [27]
B. Reed: Tree width and tangles: a new connectivity measure and some applications, in: Surveys in Combinatorics, London Math. Soc. Lecture Note Ser. 241, Cambridge Univ. Press, Cambridge, 1997, 87–162.
 [28]
N. Robertson and P. D. Seymour: Graph minors. II. Algorithmic aspects of treewidth, J. Algorithms 7 (1986), 309–322.
 [29]
N. Robertson and P. D. Seymour: Graph minors. V. Excluding a planar graph, J. Combin. Theory Ser. B 41 (1986), 92–114.
 [30]
N. Robertson and P. D. Seymour: An outline of a disjoint paths algorithm, in: Paths, Flows, and VLSILayout, B. Korte, L. Lovász, H.J. Prömel, and A. Schrijver (Eds.), SpringerVerlag, Berlin, 1990, 267–292.
 [31]
N. Robertson and P. D. Seymour: Graph minors. XIII. The disjoint paths problem, J. Combin. Theory Ser. B 63 (1995), 65–110.
 [32]
N. Robertson and P. D. Seymour: Graph minors. XXI. Graphs with unique linkages, J. Combin. Theory Ser. B 99 (2009), 583–616.
 [33]
N. Robertson and P. D. Seymour: Graph minors. XXII. Irrelevant vertices in linkage problems, J. Combin. Theory Ser. B 102 (2012), 530–563.
 [34]
N. Robertson, P. D. Seymour and R. Thomas: Quickly excluding a planar graph, J. Combin. Theory Ser. B 62 (1994), 323–348.
 [35]
A. Schrijver: Combinatorial Optimization: Polyhedra and Efficiency, SpringerVerlag, 2003.
 [36]
P. D. Seymour: Disjoint paths in graphs, Discrete Math. 29 (1980), 293–309.
 [37]
P. D. Seymour: On odd cuts and plane multicommodityows, Proceedings of the London Mathematical Society 42 (1981), 178–192.
 [38]
C. Thomassen: 2linked graph, European Journal of Combinatorics 1 (1980), 371–378.
 [39]
C. Thomassen: A simpler proof of the excluded minor theorem for higher surfaces, J. Combin. Theory Ser. B 70 (1997), 306–311.
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An extended abstract of this paper appears in SODA 2010 [14].
Research partly supported by Japan Society for the Promotion of Science, GrantinAid for Scientific Research, by C & C Foundation, by Kayamori Foundation and by Inoue Research Award for Young Scientists.
JST, ERATO, Kawarabayashi Large Graph Project, Japan.
Supported by the GrantinAid for Scientific Research, MEXT, Japan.
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Kawarabayashi, Ki., Kobayashi, Y. The edgedisjoint paths problem in Eulerian graphs and 4edgeconnected graphs. Combinatorica 35, 477–495 (2015). https://doi.org/10.1007/s0049301428286
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Mathematics Subject Classification (2000)
 05C38
 05C83
 05C85